Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras
The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters d and ℓ. The aim of the present work is to investigate the lowest weight representations of CGA with d=1 for any integer value of ℓ. First we focus on the reducibility of the Verma modules. We give a formu...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2015 |
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Інститут математики НАН України
2015
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/146864 |
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| Cite this: | Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras / N. Aizawa, R. Chandrashekar, J. Segar // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 39 назв. — англ. |
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Aizawa, N. Chandrashekar, R. Segar, J. 2019-02-11T18:07:47Z 2019-02-11T18:07:47Z 2015 Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras / N. Aizawa, R. Chandrashekar, J. Segar // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 39 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B10; 58J70 DOI:10.3842/SIGMA.2015.002 https://nasplib.isofts.kiev.ua/handle/123456789/146864 The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters d and ℓ. The aim of the present work is to investigate the lowest weight representations of CGA with d=1 for any integer value of ℓ. First we focus on the reducibility of the Verma modules. We give a formula for the Shapovalov determinant and it follows that the Verma module is irreducible if ℓ=1 and the lowest weight is nonvanishing. We prove that the Verma modules contain many singular vectors, i.e., they are reducible when ℓ≠1. Using the singular vectors, hierarchies of partial differential equations defined on the group manifold are derived. The differential equations are invariant under the kinematical transformation generated by CGA. Finally we construct irreducible lowest weight modules obtained from the reducible Verma modules. The work of N.A. is supported by the grants-in-aid from JSPS (Contract No.26400209). J.S. acknowledges the hospitality of OPU, where part of this work was completed. R.C. was financially supported through the MOST grants 102-2811-M-005-025 and 102-2628-M-005-001-MY4 in Taiwan. He would like to thank Professor Naruhiko Aizawa for the invitation to visit Osaka Prefecture University and also for the hospitality extended to him during his stay. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras |
| spellingShingle |
Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras Aizawa, N. Chandrashekar, R. Segar, J. |
| title_short |
Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras |
| title_full |
Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras |
| title_fullStr |
Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras |
| title_full_unstemmed |
Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras |
| title_sort |
lowest weight representations, singular vectors and invariant equations for a class of conformal galilei algebras |
| author |
Aizawa, N. Chandrashekar, R. Segar, J. |
| author_facet |
Aizawa, N. Chandrashekar, R. Segar, J. |
| publishDate |
2015 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters d and ℓ. The aim of the present work is to investigate the lowest weight representations of CGA with d=1 for any integer value of ℓ. First we focus on the reducibility of the Verma modules. We give a formula for the Shapovalov determinant and it follows that the Verma module is irreducible if ℓ=1 and the lowest weight is nonvanishing. We prove that the Verma modules contain many singular vectors, i.e., they are reducible when ℓ≠1. Using the singular vectors, hierarchies of partial differential equations defined on the group manifold are derived. The differential equations are invariant under the kinematical transformation generated by CGA. Finally we construct irreducible lowest weight modules obtained from the reducible Verma modules.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/146864 |
| citation_txt |
Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras / N. Aizawa, R. Chandrashekar, J. Segar // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 39 назв. — англ. |
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2025-11-28T01:22:48Z |
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2025-11-28T01:22:48Z |
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